The system in Problem 17, like the system in (2), can be solved with no advanced knowledge. Solve for x(t) and y(t) and compare their graphs with your sketches in Problem 17. Determine the limiting values of x(t) and y(t) as Explain why the answer to the last question makes intuitive sense.
(reference problem 17)
Concentration of a Nutrient Suppose compartments A and B shown in Figure 3.3.10 are filled with fluids and are separated by a permeable membrane. The figure is a compartmental representation of the exterior and interior of a cell. Suppose, too, that a nutrient necessary for cell growth passes through the membrane. A model for the concentrations x(t) and y(t) of the nutrient in compartments A and B, respectively, at time t is given by the linear system of differential equations
where VA and VB are the volumes of the compartments, and k > 0 is a permeability factor. Let x(0) = x0 and y(0) = y0 denote the initial concentrations of the nutrient. Solely on the basis of the equations in the system and the assumption x0 > y0 > 0, sketch, on the same set of coordinate axes, possible solution curves of the system. Explain your reasoning. Discuss the behavior of the solutions over a long period of time.
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